Let $M$ be a smooth $m$-manifold and $N_1$, $N_2$ smooth embedded $k$-submanifolds such that
- $N_1\cap N_2=\partial N_1=\partial N_2$,
- for each $x\in N_1\cap N_2$, $T_x N_1=T_x N_2$, and
- for $x\in N_1\cap N_2$, each vector in $T_x N_1$ is inwards-pointing wrt. $N_1$ iff it is outwards pointing wrt. $N_2$ and vice versa.
Is it true that $N_1\cup N_2$ is a smooth embedded $k$-submanifold with $N_1\cap N_2$ in its interior?
Edit: It follows from mollyerin's answer that $N$ is not necessarily the image of a $C^\infty$ embedding. But I have further bounty questions:
- Is it at least the image of a $C^1$ embedding?
- Are there some further simple conditions that I should include so that $N$ is a smooth ($C^\infty$) embedded submanifold?
- Can I "change" $N$ in some arbitrary small neighborhood of $N_1\cap N_2$ to make the resulting manifold smooth?